Imaging defects in two-dimensional crystals by convergent beam electron diffraction

arXiv:2610.21542 · cond-mat.mes-hall, physics.comp-ph, physics.optics · Submitted 2026-10-15 · Read on arXiv

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Introduction to the show: ident: Quantum Radio. Generated commentary on the latest quantum physics and condensed matter papers.

Kai: I'm Kai.

Mira: And I'm Mira, here with Lev.

Kai: Today's paper: "Imaging defects in two-dimensional crystals by convergent beam electron diffraction".

Mira: The gist Convergent beam electron diffraction (CBED) allows for atomic-precision imaging of individual defects in two-dimensional materials using one single-shot intensity measurement,

Kai: First, who's behind it and why it matters.

Title and authors: Kai: Let's talk about who wrote this and what their focus is. Tatiana Latychevskaia, Pengru Huang, Kostya Novoselov on the author list for "Imaging defects in two-dimensional crystals by convergent beam electron diffraction".

Mira: Pengru Huang is a big name in this field because he’s working on those materials, and Kai, he’s focused on how we can use these diffraction techniques to see what's happening at the atomic level.

Lev: I remember seeing some work related to these types of patterns before; it’s all about using diffraction to map out structure.

Kai: And Mira, the title itself suggests they are focusing on defects in two-dimensional crystals, which is a really important area because those materials have such unique electronic properties.

Mira: They’re not just looking at the big picture; they’re diving into how specific atomic arrangements cause patterns in the diffraction data that we can then use to identify those defects directly.

Lev: It sounds like they are bridging the gap between theoretical predictions and actual experimental observation of atomic structures.

Kai: Right, so it’s about using these simulated patterns as a guide for what you actually expect to see when you run the experiment.

The paper's summary: Mira: So, what does the paper actually show in terms of their method? They demonstrate that CBED patterns and specific constraints allow them to extract atomic information from the CBED images, which lets them image individual defects in graphene.

Kai: They use a conventional electron microscope in this mode to get these "fingerprint" patterns for individual defects in graphene, and they state this high-quality recovery of real-space information eliminates the need for those expensive aberration-corrected optics or scanning procedures.

Lev: That’s huge because if you can do that without the high cost of those lenses, it makes the technique much more practical for actual use on real hardware.

Mira: They generate simulated CBED patterns for various defects in graphene, and these simulations are based on density functional theory calculations that gave them the equilibrium atomic geometries of those defects.

Kai: So, they’re using DFT to predict the atom positions first, then simulating what the resulting CBED pattern should look like for a specific defect.

Lev: And they calculate this contrast using a specific formula, which is almost zero for the zero-order CBED spot but increases linearly with the order for higher-order spots. That’s how they tell them apart.

The paper's improvements: Kai: Now, let’s look at what they suggest as improvements to this approach. They focus on reconstructing the atomic structure of a sample from its CBED pattern by applying iterative phase retrieval methods, provided you have the right geometric conditions for correct sampling.

Mira: They lay out these steps: first, relating pixel sizes in the sample plane to pixel sizes in the detector plane via the condition zero k = two pi / N, where N is the number of pixels <ref:2610.21542#pg2>.

Lev: Then they mention a second constraint: the defocus distance, f, must be selected such that it keeps the probed region less than or equal to twice the reconstructed area to fulfill an oversampling condition.

Kai: And finally, they use iterative phase retrieval algorithms, and they suggest modifying these by removing the central part of the CBED pattern and using a perfect lattice as an initial guess to optimize convergence.

Mira: The results they got were quite solid; after one hundred iterations, the reconstructed atomic positions perfectly matched the atomic distribution in the sample, with atomic shifts due to lattice relaxation reaching zero point two five Å.

Lev: That maximal deviation of the reconstructed coordinates from the correct coordinates was zero point one Å, and they say that deviation comes from finite sampling and pixel size rather than issues with the iterative phase retrieval itself.

Conclusion: Kai: So, to wrap up on this paper about "Imaging defects in two-dimensional crystals by convergent beam electron diffraction," it seems the main point is that this method lets us get atomic-precision imaging of individual defects using just one intensity measurement.

Mira: And they show how combining DFT with CBED simulations gives us those unique defect fingerprints, which then lets us reconstruct the atomic positions of the sample through iterative phase retrieval.

Lev: The ability to map out atomic shifts, they found that the sensitivity for an out-of-plane shift is much lower than for an in-plane shift. They even quantify phase shifts: a phase shift of zero point one rad can be picked up by the interference pattern in a CBED disc, which is created by changing the defocus value.

Kai: So, what this means for us on the experimental side is that we have a clear path to imaging defects without needing super expensive optics or complex scanning setups if we use these CBED techniques.

Mira: It also gives us a way to precisely map out atomic shifts, which could be used to understand strain and displacement in these 2D materials better than before <ref:2610.21542#pg1>.

Lev: For real hardware, the point is that even with the limitations they mentioned, like finite sampling causing a zero point one Å deviation, it shows that we can recover structures that were otherwise inaccessible.

Kai: So we’ve seen how this paper on "Imaging defects in two-dimensional crystals by convergent beam electron diffraction" uses simulation and iterative reconstruction to achieve atomic precision in defect imaging.

Tatiana Latychevskaia, Pengru Huang, Kostya S. Novoselov

Paul Scherrer Institute · Institute for Functional Intelligent Materials National University of Singapore Centre for Advanced 2D Materials National University of Singapore Chongqing 2D Materials Institute National Graphene Institute University of Manchester

cond-mat.mes-hall, physics.comp-ph, physics.optics

Submitted: 2026-10-15

Updated: 2026-10-15

Journal ref: Phys. Rev. B 105, 184113, 2022

DOI: 10.1103/PhysRevB.105.184113

License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/

Importance score: 78/100

The gist: The gist Convergent beam electron diffraction (CBED) allows for atomic-precision imaging of individual defects in two-dimensional materials using one single-shot intensity measurement, which

Key concepts

Convergent Beam Electron Diffraction (CBED)
A mode in electron microscopy where a convergent beam interacts with a crystal to produce diffraction patterns. This technique is used here to obtain 'fingerprint' patterns specific to individual defects, allowing for atomic-level characterization without needing complex scanning or aberration-corrected optics.
Defect Fingerprints
Unique characteristic intensity distributions within the CBED pattern that correspond specifically to a particular defect structure in graphene. These patterns are derived from Density Functional Theory (DFT) calculations of equilibrium geometries, serving as a distinct signature for identifying and distinguishing different types of defects.
Iterative Phase Retrieval
A computational method used to reconstruct the 3D atomic structure from the 2D CBED pattern data. This process involves repeatedly propagating wavefronts between simulated sample and detector planes, using Fourier transforms, to iteratively refine an initial guess into a highly accurate atomic arrangement.
Atomic Shift Mapping
The ability of CBED interference patterns to reveal precise displacements of atoms, distinguishing between in-plane and out-of-plane distortions. Phase shifts in the pattern are sensitive indicators that can quantify atomic movements, such as those caused by lattice relaxation.

Summary

The gist

Convergent beam electron diffraction (CBED) allows for atomic-precision imaging of individual defects in two-dimensional materials using one single-shot intensity measurement, which eliminates the need to use expensive aberration-corrected optics or scanning procedures<ref:2610.21542#pg8>.

How it works

The study demonstrates that CBED patterns and particular constraints allow for the extraction of atomic information from CBED images, enabling the imaging of individual defects in 2D materials<ref:2610.21542#pg8>. The technique utilizes a conventional electron microscope operating in convergent beam electron diffraction (CBED) mode to obtain fingerprint CBED patterns of individual defects in graphene<ref:2610.21542#pg8>. This high-quality recovery of real-space information about the object eliminates the need to use expensive aberration-corrected optics or scanning procedures<ref:2610.21542#pg8>.

Simulated CBED Patterns and Defect Fingerprints

The researchers present simulated CBED patterns for various defects in graphene, each exhibiting a unique fingerprint distribution<ref:2610.21542#pg8>. These simulations were based on DFT calculations that provided the equilibrium atomic geometries of the defects<ref:2610.21542#pg10>. The contrast of the interference patterns in individual CBED spots is calculated using a specific formula, which is almost zero for the zero-order CBED spot and increases linearly with the order for higher-order spots<ref:2610.21542#pg8>. This contrast is sufficient for recognition of a characteristic interference pattern due to a defect already in a first-order CBED disc<ref:2610.21542#pg8>.

Reconstruction of Atomic Positions

The atomic structure of a sample can be reconstructed from its CBED pattern by applying iterative phase retrieval methods, provided that the patterns are acquired under geometric conditions that allow for correct sampling<ref:2610.21542#pg9>. The reconstruction process involves several steps:

  1. The pixel sizes in the sample plane (0) and in the detector (CBED pattern) plane (k) are related by the condition 0k = 2/N, where N × N is the number of pixels<ref:2610.21542#pg9>.

  2. The defocus distance f must be selected such that the probed region is less than or equal to twice the reconstructed area in order to fulfil the oversampling condition<ref:2610.21542#pg9>.

  3. Iterative phase retrieval algorithms are used, which can be modified by removing the central part of the CBED pattern and using a perfect lattice as an initial guess to optimize convergence<ref:2610.21542#pg10>.

Results of Reconstruction

The reconstructed atomic positions perfectly match the atomic distribution in the sample after 100 iterations, with atomic shifts due to lattice relaxation reaching 0.25 Å<ref:2610.21542#pg8>. The maximal deviation of the reconstructed coordinates from the correct coordinates is 0.1 Å, which originates from finite sampling and pixel size rather than the iterative phase retrieval itself<ref:2610.21542#pg9>.

Discussion on Atomic Shifts

The sensitivity of the contrast in CBED interference pattern is much lower for an out-of-plane atomic shift than for an in-plane shift<ref:2610.21542#pg9>. Phase shifts for out-of-plane shift z and in-plane shift x are given by specific formulas involving sin terms related to the phase angle<ref:2610.21542#pg10>. A phase shift of 0.1 rad can be picked up by the interference pattern in a CBED disc, which can be created by a change in defocus value<ref:2610.21542#pg8>. The extent of the overlap between complex-valued waves can be selected by choosing the size of the imaged area, which is regulated by regulating the defocus distance<ref:2610.21542#pg10>.

Methods Used

The equilibrium geometries of defect structures were obtained via DFT calculations using the PBE functional implemented in VASP 5.4<ref:2610.21542#pg10>. The transmission function of a monolayer was calculated by convolving the projected potential of an individual atom with the function describing positions of the atoms in the lattice<ref:2610.21542#pg10>. The CBED pattern is simulated as the square of the amplitude of the Fourier transform of the exit wave<ref:2610.21542#pg10>. The iterative phase retrieval was performed by performing wavefront propagation back and forth between the sample and detector planes, using an FT and inverse FT, respectively<ref:2610.21542#pg10>.

Author Contributions

T. L. and K. N. initiated the research and analysed the data, P. H. designed the models and performed the DFT simulations, T. L. performed the diffraction simulations, all authors discussed the results and contributed to the final manuscript<ref:2610.21542#pg10>.

Acknowledgements

The authors acknowledge support by various funding bodies including the Ministry of Education (Singapore) through its Research Centre of Excellence program, EU Graphene Flagship Program, European Research Council Synergy Grant Hetero2D, and others<ref:2610.21542#pg10>.

References

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Improvements for AI systems

  1. textbf Citrine Defect Fingerprinting for Material Identification (CBED Analysis)

This system can perform atomic-precision imaging of individual defects in 2D materials using one single-shot intensity measurement, allowing for the direct assignment of a defect by its unique fingerprint CBED pattern, as demonstrated by the contrast calculations: "The contrast of the interference patterns in the individual CBED spots, calculated as C I I I I = - + (max min max min) /, is almost zero for the zero-order CBED spot, and increases linearly with the order for the higher-order CBED spots."

  1. textbf Automated Atomic Position Reconstruction

The system can reconstruct atomic positions, including the individual atomic defects in graphene, by iterative phase retrieval from a single CBED pattern, achieving a maximal deviation of the reconstructed coordinates from the correct coordinates is 0.1 Å when using specific constraints, effectively recovering structures that are otherwise inaccessible to conventional microscopy.

  1. textbf Strain and Displacement Mapping

The AI can quantify atomic shifts, specifically differentiating between in-plane and out-of-plane distortions by utilizing the phase shifts derived from CBED interference patterns: A phase shift of 0.1 rad can be easily picked up by the interference pattern in a CBED disc, and can be created by Δz = 0.35 nm or Δx = 3.39 pm.

  1. textbf Robust Phase Retrieval Algorithm

The AI system will incorporate a modified iterative phase retrieval algorithm optimized for experimental conditions, featuring removing the central part of the CBED pattern, and by using a perfect lattice as an initial guess, ensuring high stability and quick convergence even for noisy (SNR = 10) data.

  1. textbf Defect Structure Prediction via DFT Integration

The system can utilize structural calculations using density functional theory to provide inputs for CBED simulations, allowing it to generate simulated patterns based on predicted equilibrium geometries: In this study, the equilibrium atomic geometries of the defects, used as inputs for the CBED simulations, were obtained from DFT calculations.

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